Can you please provide a reference to the following generalization of Vandermonde's identity?
Given a positive integer $k$ and nonnegative integers $n_1, n_2, \ldots, n_k$ and $m$, it holds that $$\sum_{i_1+i_2+\cdots+i_k=m} \binom{n_1}{i_1} \binom{n_2}{i_2} \cdots \binom{n_k}{i_k} = \binom{n_1+n_2+\cdots+n_k}{m}.$$ The proof is well-known and based on the idea of counting in two different ways the coefficient of $x^m$ in the polynomial $(1+x)^{n_1+n_2+\cdots+n_k}$.
The generalised identity is stated without proof in Concrete Mathematics, exercise 5.62. The proof technique that you mention is used for the ungeneralised Vandermonde identity in section 5.4.
\begin{array}{l} \sum\limits_{i_1 + i_2 + \cdots + i_k = m} {\left( \begin{array}{c} n_1 \\ i_1 \\ \end{array} \right)\left( \begin{array}{c} n_2 \\ i_2 \\ \end{array} \right)\left( \begin{array}{c} n_3 \\ i_3 \\ \end{array} \right) \cdots \left( \begin{array}{c} n_k \\ i_k \\ \end{array} \right)} = \\ = \sum\limits_{i_1 + i_2 = m - \left( {i_3 \cdots + i_k } \right)} {\left( {\left( \begin{array}{c} n_1 \\ i_1 \\ \end{array} \right)\left( \begin{array}{c} n_2 \\ i_2 \\ \end{array} \right)} \right)\left( \begin{array}{c} n_3 \\ i_3 \\ \end{array} \right) \cdots \left( \begin{array}{c} n_k \\ i_k \\ \end{array} \right)} = \\ = \sum\limits_{i_3 = m - \left( {i_4 \cdots + i_k } \right)} {\left( {\left( \begin{array}{c} n_1 + n_2 \\ m - \left( {i_3 \cdots + i_k } \right) \\ \end{array} \right)} \right)\left( \begin{array}{c} n_3 \\ i_3 \\ \end{array} \right) \cdots \left( \begin{array}{c} n_k \\ i_k \\ \end{array} \right)} = \\ = \cdots \\ \end{array}
